MCTS: How to choose the final action from the root
By far the most commonly used strategy is to select the child with the highest number of visits. This is as described in the 2008 paper you linked. It's also what's referred to as the "robust child" in the 2012 paper you linked.

In algorithm 2 of the 2012 paper, they actually use the highest average reward, which corresponds to "Max child". It looks like they're using the UCB1 policy, but they actually use a value of $0$ for the exploration parameter $c$, which makes the entire square root term drop out. This is also explained in the text at the end of your quote. But usually, a robust child / max visit count performs better.

Progressive Strategies for Monte-Carlo Tree Search is a different paper from 2008, in which these strategies are experimented with a bit. Usually, they perform similarly, but a robust child tends to be the best if there is any difference at all.